\[ \int \frac {-4 x^2+\left (5-e^5+3 x\right ) \log (15)}{\left (10 x^2-2 e^5 x^2+2 x^3+\left (-5 x+e^5 x-x^2\right ) \log (15)\right ) \log \left (\frac {-2 x+\log (15)}{e^{15} x+e^{10} \left (-10 x-2 x^2\right )+e^5 \left (25 x+10 x^2+x^3\right )}\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (\frac {\left (\frac {\ln \left (15\right )}{x}-2\right ) {\mathrm e}^{-5}}{\left ({\mathrm e}^{5}-x -5\right )^{2}}\right )\right ) \]
command
Int[(-4*x^2 + (5 - E^5 + 3*x)*Log[15])/((10*x^2 - 2*E^5*x^2 + 2*x^3 + (-5*x + E^5*x - x^2)*Log[15])*Log[(-2*x + Log[15])/(E^15*x + E^10*(-10*x - 2*x^2) + E^5*(25*x + 10*x^2 + x^3))]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-4 x^2+\left (5-e^5+3 x\right ) \log (15)}{\left (10 x^2-2 e^5 x^2+2 x^3+\left (-5 x+e^5 x-x^2\right ) \log (15)\right ) \log \left (\frac {-2 x+\log (15)}{e^{15} x+e^{10} \left (-10 x-2 x^2\right )+e^5 \left (25 x+10 x^2+x^3\right )}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (\log \left (-\frac {2 x-\log (15)}{e^5 x \left (x-e^5+5\right )^2}\right )\right ) \]